Self-collimation and focusing effects in zero-average index metamaterials

Rémi Pollès; Emmanuel Centeno; Julien Arlandis; Antoine Moreau

doi:10.1364/OE.19.006149

Self-collimation and focusing effects in zero-average index metamaterials

Rémi Pollès, Emmanuel Centeno, Julien Arlandis, and Antoine Moreau

Optics Express
Vol. 19,
Issue 7,
pp. 6149-6154
(2011)
•https://doi.org/10.1364/OE.19.006149

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Rémi Pollès, Emmanuel Centeno, Julien Arlandis, and Antoine Moreau, "Self-collimation and focusing effects in zero-average index metamaterials," Opt. Express 19, 6149-6154 (2011)

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Related Topics
Table of Contents Category
- Metamaterials
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Metamaterials (160.3918)
- Photonic bandgap materials (160.5293)

About this Article
History
- Original Manuscript: January 21, 2011
- Revised Manuscript: February 22, 2011
- Manuscript Accepted: February 22, 2011
- Published: March 17, 2011

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Open Access

Abstract

One dimensional photonic crystals combining positive and negative index layers have shown to present a photonic band gap insensitive to the period scaling when the volume average index vanishes. Defect modes lying in this zero-n̄ gap can in addition be obtained without locally breaking the symmetry of the crystal lattice. In this work, index dispersion is shown to broaden the resonant frequencies creating then a conduction band lying inside the zero-n̄ gap. Self-collimation and focusing effects are in addition demonstrated in zero-average index metamaterials supporting defect modes. This beam shaping is explained in the framework of a beam propagation model by introducing an harmonic average index parameter.

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References

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|
Year
|
Author
|
Publication

J. Joannopoulos, S. Johnson, D. Winn, and R. Meade, Photonic crystals: molding the flow of light, 2nd edition, (Princeton University Press2008).
I. Nefedov and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611 (2002).
[Crossref]
J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[Crossref]
D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[Crossref]
Y. Yuan, L. Ran, J. Huangfu, H. Chen, L. Shen, and J. Kong, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220–2227 (2006).
[Crossref] [PubMed]
S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[Crossref] [PubMed]
V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]
N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[Crossref]
I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[Crossref]
F. Krayzel, R. Pollès, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc. : Rap. Pub. 5, 10025 (2010).
[Crossref]
E. Silvestre, R. Depine, M. Martínez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
[Crossref]
H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]
Throughout this paper, the inverse Fourier Transform is defined by TF−1(f(α))=∫−∞∞f(x)exp(iαx)dα.
M. Born, E. Wolf, and A. B. Bhatia, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (Cambridge University press, Cambridge2000).
[PubMed]

2010 (1)

F. Krayzel, R. Pollès, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc. : Rap. Pub. 5, 10025 (2010).
[Crossref]

2009 (4)

E. Silvestre, R. Depine, M. Martínez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
[Crossref]

S. Kocaman, R. Chatterjee, N. Panoiu, J. Mcmillan, M. Yu, R. Osgood, D. Kwong, and C. Wong, “Observation of zeroth-order band gaps in negative-refraction photonic crystal superlattices at near-infrared frequencies,” Phys. Rev. Lett. 102, 203905 (2009).
[Crossref] [PubMed]

V. Mocella, S. Cabrini, A. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102, 133902 (2009).
[Crossref] [PubMed]

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[Crossref]

2006 (2)

N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[Crossref]

Y. Yuan, L. Ran, J. Huangfu, H. Chen, L. Shen, and J. Kong, “Experimental verification of zero order bandgap in a layered stack of left-handed and right-handed materials,” Opt. Express 14, 2220–2227 (2006).
[Crossref] [PubMed]

2004 (1)

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. Vigneron, E. E. Boudouti, and A. Nougaoui, “Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials,” Phys. Rev. E 69, 066613 (2004).
[Crossref]

2003 (1)

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[Crossref]

2002 (1)

I. Nefedov and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611 (2002).
[Crossref]

1999 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

Akjouj, A.

Bhatia, A. B.

M. Born, E. Wolf, and A. B. Bhatia, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (Cambridge University press, Cambridge2000).
[PubMed]

Born, M.

M. Born, E. Wolf, and A. B. Bhatia, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (Cambridge University press, Cambridge2000).
[PubMed]

Boudouti, E. E.

Bria, D.

Brueck, S.

N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[Crossref]

Cabrini, S.

Chan, C.

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[Crossref]

Chang, A.

Chatterjee, R.

Chen, H.

Dardano, P.

Depine, R.

E. Silvestre, R. Depine, M. Martínez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
[Crossref]

Dhuey, S.

Djafari-Rouhani, B.

Dobrzynski, L.

Harteneck, B.

Huangfu, J.

Joannopoulos, J.

J. Joannopoulos, S. Johnson, D. Winn, and R. Meade, Photonic crystals: molding the flow of light, 2nd edition, (Princeton University Press2008).

Johnson, S.

J. Joannopoulos, S. Johnson, D. Winn, and R. Meade, Photonic crystals: molding the flow of light, 2nd edition, (Princeton University Press2008).

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

Kivshar, Y.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[Crossref]

Kocaman, S.

Kong, J.

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

Krayzel, F.

F. Krayzel, R. Pollès, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc. : Rap. Pub. 5, 10025 (2010).
[Crossref]

Kwong, D.

Li, J.

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[Crossref]

Martínez-Ricci, M.

E. Silvestre, R. Depine, M. Martínez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
[Crossref]

Mcmillan, J.

Meade, R.

J. Joannopoulos, S. Johnson, D. Winn, and R. Meade, Photonic crystals: molding the flow of light, 2nd edition, (Princeton University Press2008).

Mihailovic, M.

F. Krayzel, R. Pollès, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc. : Rap. Pub. 5, 10025 (2010).
[Crossref]

Mocella, V.

Monsoriu, J.

E. Silvestre, R. Depine, M. Martínez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
[Crossref]

Moreau, A.

F. Krayzel, R. Pollès, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc. : Rap. Pub. 5, 10025 (2010).
[Crossref]

Moretti, L.

Nefedov, I.

I. Nefedov and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611 (2002).
[Crossref]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

Nougaoui, A.

Olynick, D.

Osgood, J.

N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[Crossref]

Osgood, R.

Panoiu, N.

N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[Crossref]

Pollès, R.

F. Krayzel, R. Pollès, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc. : Rap. Pub. 5, 10025 (2010).
[Crossref]

Ran, L.

Rendina, I.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

Shadrivov, I.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[Crossref]

Shen, L.

Sheng, P.

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[Crossref]

Silvestre, E.

E. Silvestre, R. Depine, M. Martínez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
[Crossref]

Sukhorukov, A.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[Crossref]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

Tretyakov, S.

I. Nefedov and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611 (2002).
[Crossref]

Vigneron, J.

Winn, D.

J. Joannopoulos, S. Johnson, D. Winn, and R. Meade, Photonic crystals: molding the flow of light, 2nd edition, (Princeton University Press2008).

Wolf, E.

M. Born, E. Wolf, and A. B. Bhatia, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (Cambridge University press, Cambridge2000).
[PubMed]

Wong, C.

Yu, M.

Yuan, Y.

Zhang, S.

N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[Crossref]

Zhou, L.

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[Crossref]

Appl. Phys. Lett. (2)

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2009).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

J. Europ. Opt. Soc. : Rap. Pub. (1)

F. Krayzel, R. Pollès, A. Moreau, and M. Mihailovic, “Simulation and analysis of exotic non-specular phenomena,” J. Europ. Opt. Soc. : Rap. Pub. 5, 10025 (2010).
[Crossref]

J. Opt. Soc. Am. B (2)

E. Silvestre, R. Depine, M. Martínez-Ricci, and J. Monsoriu, “Role of dispersion on zero-average-index bandgaps,” J. Opt. Soc. Am. B 26, 581–586 (2009).
[Crossref]

N. Panoiu, J. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506–513 (2006).
[Crossref]

Opt. Express (1)

Phys. Rev. E (2)

I. Nefedov and S. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611 (2002).
[Crossref]

Phys. Rev. Lett. (3)

J. Li, L. Zhou, C. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 83901 (2003).
[Crossref]

Other (3)

J. Joannopoulos, S. Johnson, D. Winn, and R. Meade, Photonic crystals: molding the flow of light, 2nd edition, (Princeton University Press2008).

Throughout this paper, the inverse Fourier Transform is defined by TF−1(f(α))=∫−∞∞f(x)exp(iαx)dα.

M. Born, E. Wolf, and A. B. Bhatia, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light (Cambridge University press, Cambridge2000).
[PubMed]

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Fig. 1

(a) and (c): dispersion relation given by Eq. (1) (black curve) and its Taylor expansions (Eq. (2)) (red curve). (b) and (d): reflectance of the structure composed of N = 50 periods. Black curves correspond to the nondispersive case and red curves to the dispersive case. (a) and (b): the structure is characterized by n₁ = 1, $n_{2}^{0} = - 2$ , η₁ = 1, η₂ = 0.5, d₁ = 2D/3, d₂ = D/3 and D/λ₀ = 13/8 (which corresponds to D = 1.625λ₀). (c) and (d): the parameters are the same but D/λ₀ = 3/2.

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Fig. 2

Modulus of the field when a PBGM, embedded in a air-medium (n₀ = 1), is illuminated by a beam. (a) The parameters of the PBGM are N = 200, d₁ = d₂ = D/2, n₁ = 2, $n_{2}^{0} = - 2$ , η₁ = 1, η₂ = 0.5 and D = λ₀. (b) The parameters of the PBGM are N = 400, d₁ = D/3, d₂ = 2D/3, n₁ = 1, $n_{0}^{2} = - 0.5$ , η₁ = 1, η₂ = 0.5 and D = 3λ₀.

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Equations (11)

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cos (κ D) = cos (\bar{n} k D) + \frac{{(η_{1} - η_{2})}^{2}}{2 η_{1} η_{2}} sin (n_{1} k d_{1}) sin (| n_{2} | k d_{2}),

cos (κ D) ≃ 1 + Γ x_{m}^{2} + Γ Δ n_{2} (λ) k d_{2} x_{m} - {(Δ n_{2} (λ) k d_{2})}^{2} / 2,

Δ_{m} λ = \frac{m A (η_{2}^{2} - η_{1}^{2}) (λ_{0} - Λ_{m})}{A^{2} η_{2} η_{1} - m (A + m) {(η_{2} - η_{1})}^{2}},

U (x, L) = T F^{- 1} {U^{i} (α) {(P_{1} (α, d_{1}) P_{2} (α, d_{2}))}^{N}} .

U (x, L) = T F^{- 1} {U^{i} (α) \tilde{P} {(α, D)}^{N}} .

U (x, L) = \frac{W_{0}}{\bar{ω} (L)} e^{- {(\frac{x}{W (L)})}^{2}} e^{i \frac{ω}{c} 〈 n 〉 L} e^{i φ (x, L)},

W (L) = W_{0} \sqrt{1 + θ_{0}^{2} {(N D)}^{2} {〈 \frac{1}{n} 〉}^{2}},

\frac{d_{1}}{n_{1}} + \frac{d_{2}}{n_{2}} = 0 .

U (x, L) = T F^{- 1} {U^{i} (α) P_{0} (α, f) \tilde{P} {(α, D)}^{N} P_{0} (α, f^{'})},

W (f^{'}) = W_{0} \sqrt{1 + θ_{0}^{2} {(\frac{f}{n_{0}} + N D 〈 \frac{1}{n} 〉 + \frac{f^{'}}{n_{0}})}^{2}} .

f + f^{'} = - N D 〈 \frac{1}{n} 〉 n_{0} .